Accurate prediction of flow behaviors in reservoirs is essential for effective reservoir management. Reservoir simulation is the use of a physical model of a reservoir on a computer to test how the reservoir will perform as production or stimulation proceeds over time, i.e. over a series of discrete time steps. The reservoir model is spatially discretized so that the differential equations governing fluid flow can be solved by numerical methods. Discretization methods commonly used today for reservoir simulation can be generally classified into one of two categories: finite difference/finite volume methods (FDM) or finite element methods (FEM). Of the different choices, each numerical method for solving differential equations on a computer has advantages and disadvantages. It is well-known that a finite difference method typically is simpler, easier to implement, and executes faster. For this reason, finite difference/finite volume methods are used widely in the industry today. Unfortunately, finite difference methods such as two-point flux approximation (TPFA) require Voronoi grids, where each cell boundary is perpendicular to the line joining centers of the two neighboring cells; see, for example, Heinrich. This is a severe limitation because building a Voronoi grid is challenging if not impossible when a reservoir model contains intersecting faults, pinchouts, or other irregular geological features. Finite element methods, on the other hand, are more complex mathematically, more difficult to implement, and take longer to execute. Because of this, finite element methods are not commonly used in reservoir simulation, even though they have the advantage of being applicable to models with flexible grids and using a general permeability tensor. Because of the implementation and efficiency issues associated with finite element and other alternatives, finite difference/finite volume methods such as TPFA are used in practice sometimes in situations where they should not, throwing into doubt the validity of simulation results.
While extensive studies exist in literature on mathematical theory of finite difference/finite volume methods for reservoir simulation, papers published on application of finite element methods to modeling general multiphase (gas or liquid) flow are limited. Most of the publications related to FEM methods have made simplifications either on fluid phase behavior, or simplifying treatments on gravitational or capillary effect. For example, Cai et al. proposed a new control-volume mixed finite element method for irregular block-centered quadrilateral grids and tested it on single phase flow problems. Hoteit et al. (2002) carried out analysis of mixed finite element and mixed-hybrid finite element methods for single phase problems. For multiphase flow, Fung et al. developed a control-volume finite-element method (CVFEM) for flow simulation to reduce grid-orientation effects. For CVFEM, mass conservation is honored on the dual grid. By comparison, a mixed finite element method (MFEM) offers a more natural way for achieving mass conservation, which is satisfied on the original grid. Chavent et al. used MFEM to simulate incompressible two-phase flow in two dimensions. MFEM was also used by Darlow et al. for solving miscible displacement problems and by Ewing and Heinemann for performing compositional simulation while neglecting gravity and capillary pressure. To improve accuracy, Durlofsky et al. developed a mixed finite element method for modeling three phase, multicomponent systems using IMPES type formulation. A similar MFEM approach was employed by Hoteit et al. (2006) in combination with discontinuous Galerkin to capture sharp saturation gradient. For parallel multiphysics and multiscale simulation, a multiblock/multidomain approach using mixed and expanded mixed methods and mortar space to handle non-matching grids was proposed by Wheeler et al.
Besides FEM methods, the multipoint flux approximation (MPFA, see for example, Aavatsmark et al., Chen et al, Edwards et al.) has been developed to handle permeability tensor and flexible grids for reservoir simulation. It has been shown that with special choices of finite element spaces and the quadrature rule, MPFA may be derived from FEM methods.